Influence of intrinsic decoherence on entanglement degree in the atom–field coupling system

In this paper, we use the quantum mutual entropy to measure the degree of entanglement in the time development of a two-level particle (atom or trapped ion). We find an exact solution of the Milburn equation for the system. The exact solution is then used to discuss the influence of intrinsic decoherence on degree of entanglement. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. It is shown that the degree of entanglement is very sensitive to the changes of the intrinsic decoherence. The results show that the effect of the intrinsic decoherence decreases the quasiperiod of the entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.     

Entropy and entanglement of an effective two-level atom interacting with two quantized field modes in squeezed displaced fock states

We have studied the influences of ac-Stark shifts on the field quantum entropy, with “squeezed displaced Fock states” (SDFSs) basis. By a unitary transformation we derive a Raman-coupled Hamiltonian perturbatively in coupling constants. The exact results are employed to perform a careful investigation of the temporal evolution of entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in the SDFS. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The results show that the effect of the SDFS changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The Rabi oscillation frequency, the collapse and revival times of the atomic coherence are found to have strikingly different photon-intensity dependent than those found previously. The general conclusions reached are illustrated by numerical results.     

Effect of atomic decay on purity,total correlations and entanglement of pure and mixed states

The atomic decay for a two level atom interacting with a single mode of electromagnetic field is considered. In particular for a coherent state or statistical mixture (SM) of two opposite coherent states as initial field states, the exact solution of the master equation is found. Effect of the atomic damping on the partial entropies of the atom or the field and the total entropy as a measures of the purity loss is investigated. The degree of entanglement by the negativity and the mutual information and the atomic coherence through the master equation is studied.     

Entanglement,fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.     

Quantum proper entropies and additive capacities of quantum information channels

An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.     

Holographic instant conformal symmetry breaking by colliding conical defects

We study instant conformal symmetry breaking as a holographic effect of ultrarelativistic particles moving in the AdS₃ space–time. We give a qualitative picture of this effect based on calculating the two-point correlation functions and the entanglement entropy of the corresponding boundary theory. We show that in the geodesic approximation, because of gravitational lensing of the geodesics, the ultrarelativistic massless defect produces a zone structure for correlators with broken conformal invariance. At the same time, the holographic entanglement entropy also exhibits a transition to nonconformal behavior. Two colliding massless defects produce a more diverse zone structure for correlators and the entanglement entropy.     

Thermomechanical effects in the flow of a fluid in porous media

This paper deals with analysis, by methods of extended thermodynamics, of the thermomechanical effects which arise in the flow of a weakly viscous fluid in a porous medium. Under the hypothesis that the fluid fills all the interstices among the powder and that the size of the powder grains and of the interstices is much lower than a suitable characteristic length, linearized field equations are written, which include, in a natural way, terms which take into account the Dufour, Soret, and virtual mass effects. As a limiting case when the evolution time of the heat flux goes to infinite and no entropy flux is carried, the flow of liquid helium II in a porous medium is obtained.     

Multiplicity of stationary solutions to the Euler-Poisson equations

Consider the system of Euler-Poisson as a model for the time evolution of gaseous stars through the self-induced gravitational force. We study the existence, uniqueness and multiplicity of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy a priori. These results generalize the previous works on the irrotational or the rotational gaseous stars around an axis, and then they hold in more general physical settings. Under the assumption of radial symmetry, the monotonicity properties of the radius of the gas with respect to either the strength of the velocity field or the center density are also given which yield the uniqueness under some circumstances.     

On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.     

Phase-field models for fluid mixtures

The paper investigates the modelling of phase transitions in multiphase fluid mixtures. The order parameter is identified with the set of concentrations and is a phase field in that it varies smoothly in the space region. This in turn requires that the continuity equations be regarded as constraints on the pertinent fields. The phase field is viewed as an internal variable whose evolution is subject to thermodynamic requirements. The second law allows for an extra entropy flux which proves to be proportional to the time derivative of the order parameter. Previous papers on the subject are revisited. It follows that their recourse to external mass supplies or to ad-hoc entropy fluxes can be avoided. The analogy of the phase-field model, with that of mixtures with mass–density gradients and extra entropy flux, is emphasized.     

Infinite dimensional attractors for porous medium equations in heterogeneous medium

In this paper, we give a detailed study of the global attractors for porous medium equations in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is obtained by showing that their ?‐Kolmogorov entropy behaves as a polynomial of the variable 1 ∕ ? as ? tends to zero, which is not observed for non‐degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ?‐entropy of infinite‐dimensional attractors are also obtained. We believe that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

New one‐leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non‐negativity and the entropy dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G‐stability theory with entropy dissipation methods and to introduce a nonlinear transformation of variables, which provides a quadratic structure in the equations. It is shown that G‐stability of the one‐leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross‐diffusion system from population dynamics and a nonlinear fourth‐order quantum diffusion model, for which the existence of semidiscrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second‐order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1119–1149, 2015     

A representation of the equations of multicomponent multiphase seepage

A mathematical model of non-isothermal multicomponent flows in a porous medium is investigated. A general case is considered when the model can be used to describe processes with an arbitrary number of components and phases. A general form of the system of mixed-type equations describing the flow, which is similar to the Godunov form for hyperbolic systems is proposed. The equations obtained are applicable to flows with gas, liquid and solid phases. The thermodynamic properties of the medium are determined solely by a single multivalued function, by changing which one can obtain models of different flows in a porous medium. A clear geometrical interpretation of the solutions of the equations is proposed. An equation for the entropy is obtained, and it is shown that in order that the model should not contradict the second law of thermodynamics, it is necessary to take into account, in the energy equation, the work of the gravity force, which is often neglected when investigating seepage.     

The simplest equation method to study perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

The simplest equation method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations.In this paper, the simplest equation method is used to construct exact solutions of nonlinear Schrödinger’s equation and perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. It is shown that the proposed method is effective and general.     

Shock waves in the nonisentropic gas flow

In this paper we propose an extended entropy condition for general systems of hyperbolic conservation laws with several space variables. This entropy condition generalizes the well-known condition (E) of Volpert for a single conservation law with several space variables and reduces to the entropy condition proposed earlier by the author for systems with one space variable. The Riemann problem for general nonisentropic gas equations has a unique solution for initial data with arbitrarily large jumps. The occurrence of a vacuum region is observed. The projections of shock curves on the pressure-velocity plane are analyzed so as to study the interaction of weak shocks. Our results differ markedly from those of previous works in that we do not assume the equation of state to be polytropic. In fact our assumptions on the equation of state allow the pressure to be a nonconvex function of specific volume.The Riemann problem for this general system of gas equations was also treated by B. Wendroff when the initial data are near constant.     

On the uniqueness of smooth,stationary black holes in vacuum

A fundamental conjecture in general relativity asserts that the domain of outer communication of a regular, stationary, four dimensional, vacuum black hole solution is isometrically diffeomorphic to the domain of outer communication of a Kerr black hole. So far the conjecture has been resolved, by combining results of Hawking [17], Carter [4] and Robinson [28], under the additional hypothesis of non-degenerate horizons and real analyticity of the space-time. We develop a new strategy to bypass analyticity based on a tensorial characterization of the Kerr solutions, due to Mars [24], and new geometric Carleman estimates. We prove, under a technical assumption (an identity relating the Ernst potential and the Killing scalar) on the bifurcate sphere of the event horizon, that the domain of outer communication of a smooth, regular, stationary Einstein vacuum spacetime of dimension 4 is locally isometric to the domain of outer communication of a Kerr spacetime.     

Dynamics of dark–bright vector solitons in a two-component Bose–Einstein condensate

Coupled dark–bright vector solitons are considered in a two-component Bose–Einstein condensate, and their dynamics are investigated by the variational approach based on the renormalized integrals of motion. The stationary states and their atom population distribution are obtained, and it is found that the dark soliton has obvious robust features. The dynamic mechanism is demonstrated by performing a coordinate of a classical particle moving in an effective potential field, and the switching and self-trapping dynamics of the coupled dark–bright vector solitons are discussed by the evolution of the atom population transferring ratio.